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I put in a reference to Lurie’s course on Tamagawa numbers at Ran space.
And a Wikipedia link, and a reference to the ?original reference (according to Wikipedia, which isn’t terribly assertive on that point).
Good point. I suppose so.
So not quite: But the sphere spectrum comes from a “charged” Ran space, see Example 13 here
What goes wrong then? The topology?
The “charged” version sounds very like the kind of thing John Baez used to discuss on TWF. Let’s see. From TWF 151:
By the way, there’s something called the Thom-Dold theorem that lets us generalize the heck out of this. We just showed that if you take the 2-sphere and consider the space of particle-antiparticle swarms in it, you get K(Z,2). But suppose instead we started with the n-sphere and considered the space of particle-antiparticle swarms in that. Then we’d get K(Z,n)!
More generally, suppose we didn’t use integers to say how many particles were at each point in the n-sphere - suppose we used elements of some abelian group A. Then we’d get K(A,n)!
For more tricks like this, try this paper:
3) Dusa McDuff, Configuration spaces of positive and negative particles, Topology 14 (1975), 91-107.
Oh yes, we had that long conversion once on fundamental categories with duals on a stratified space (not just exit or entrance paths), where we’d be able to fill in the dots of
The category of unframed/framed, unoriented/oriented tangles in 3 dimensions is the free braided monoidal category with duals on …
(This was with the knot theorist’s ’framing’ rather than the homotopy theorist’s.)
Hmm. Is that just a coincidence that we’ve started talking here about charged particles combining, when in Part 1 of Equivariant Stable Cohomotopy and Branes you’re looking at charged branes and anti-branes coinciding?
Right, shouldn’t be a coincidence!
I wonder how seriously Graeme Segal took his gloss here on the result that there’s a map from the space of finite subsets of $\mathbb{R}^n$ to $\Omega^n S^n$. He assigns to a finite set, $c$, of unit charged particles in $\mathbb{R}^n$ a map, $E_c$, which sends a point to its electric field value there, defining it also at the points of $c$ and at infinity.
He writes “Despite its picturesqueness the electrostatic map…is not very convenient in practice.”
So I need to get to speed with these things, but this “electric field map” is supposed to be a huge deal these days. We heard about it in various talks here at Geometry, Topology and Physics 2018. Apparently Segal said smewhere that he regards this insight as his greatest contribution to Mathematics. (!)
But are people just using it as a name, or is there something to learn from the physics setting? There’s a nice looking set of notes by Knudson Configuration spaces in algebraic topology where the electric field map gets a mention (p. 49), but I don’t see that the physics is treated as telling us anything.
Don’t we learn from history that it pays to attend to such physical settings, e.g., Gauss’s linking integral for knots taken up by Witten. There are examples from Felix Klein too.
But then I hardly need to tell you that.
Right, so what seems to be going on is this:
The role of configuration spaces of, specifically, charged particles, as presentations of fundamental objects of abstract (stable) homotopy theory, is well known and at the heart of recent developments. At the same time, the relation of that to any actual physics, despite and beyond the striking resemblance, seems to have been missing. I’d subscribe to the conjecture that Equivariant Stable Cohomotopy and Branes is the missing link. Now to substantiate this, let’s dig a bit deeper. Should be fun.
Hm, so my suggestion in #5 is still not right/not what we are after: That example (here) is still not about a Ran-space of $\pm$-charged particles, but about a configuration space of particles with charges in $\{0,1\}$.
The $\pm$-version could be what McDuff’s old article (McDuff 75) is amplifying, regarding the configuration spaces $C^{\pm}$ discussed there. The relevant discussion for us would be that on the bottom of p. 6, but I am not sure yet which punchline to extract from that.
Have to quit for tonight…
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